We are tasked with finding the derivative of the function:

$y = \frac{x e^x}{\cos x}$

To differentiate this, we will use the quotient rule, which is used when differentiating a quotient of two functions, combined with the product rule for the numerator. Let's go step by step.

1. Quotient Rule:

The quotient rule states that for two functions $u(x)$ and $v(x)$, the derivative of their quotient is:

$\left( \frac{u(x)}{v(x)} \right)' = \frac{v(x) u'(x) - u(x) v'(x)}{(v(x))^2}$

In this case:

• $u(x) = x e^x$
• $v(x) = \cos x$

We'll need to compute the derivatives of $u(x)$ and $v(x)$.

2. Product Rule (for $u(x) = x e^x$):

The product rule states that for two functions $f(x)$ and $g(x)$, their product's derivative is:

$(f(x) g(x))' = f'(x) g(x) + f(x) g'(x)$

Here, for $u(x) = x e^x$, we treat:

• $f(x) = x$ and
• $g(x) = e^x$

Thus, by the product rule:

$u'(x) = \frac{d}{dx}(x) \cdot e^x + x \cdot \frac{d}{dx}(e^x)$ $u'(x) = 1 \cdot e^x + x \cdot e^x = e^x + x e^x$

3. Derivative of $v(x) = \cos x$:

We know that: $v'(x) = \frac{d}{dx}(\cos x) = -\sin x$

4. Applying the Quotient Rule:

Now that we have $u'(x) = e^x + x e^x$ and $v'(x) = -\sin x$, we apply the quotient rule:

$y' = \frac{v(x) u'(x) - u(x) v'(x)}{(v(x))^2}$

Substituting in the expressions for $u(x)$, $u'(x)$, $v(x)$, and $v'(x)$:

$y' = \frac{\cos x \cdot (e^x + x e^x) - (x e^x) \cdot (-\sin x)}{(\cos x)^2}$

Simplify the numerator:

$y' = \frac{\cos x \cdot (e^x + x e^x) + x e^x \sin x}{\cos^2 x}$

We can further simplify the terms in the numerator if needed, but this is the derivative of the given function.

Final Result:

$y' = \frac{\cos x (e^x + x e^x) + x e^x \sin x}{\cos^2 x}$

This is the derivative of the function $y = \frac{x e^x}{\cos x}$.

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Do you want any more details on a specific step or further simplification?

Here are 5 related questions you might consider:

1. How do you apply the chain rule in complex functions?
2. What is the derivative of $\frac{e^x}{\cos x}$ without the extra $x$?
3. Can we find the second derivative of this function?
4. How do you interpret the product rule geometrically?
5. How does the quotient rule differ from the product rule in terms of usage?

Tip: When working with quotient rule problems, always differentiate both numerator and denominator separately before applying the rule to avoid mistakes.